cosA-sinA+1/cosA+sinA-1=cot A+cosecA
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LHS = (cosA - sinA + 1)/(cosA + sinA - 1)
multiplying and dividing (cosA - sinA + 1)
= (cosA - sinA + 1) × (cosA - sinA + 1)/(cosA + sinA - 1 ) × (cosA - sinA + 1)
= (cosA - sinA + 1)²/{cos²A - (sinA - 1)²}
= (cos²A + sin²A - 2sinA.cosA + 1 + 2cosA - 2sinA)/{cos²A - sin²A - 1 + 2sinA}
= (1 - 2sinA.cosA + 1 + 2cosA - 2sinA)/(1 - sin²A - sin²A - 1 + 2sinA)
= (2 - 2sinA + 2cosA - 2sinA.cosA)/(2sinA - 2sin²A)
= {2(1 - sinA) + 2cosA(1 - sinA)}/{2sinA(1 - sinA)}
= 2(1 + cosA)(1 - sinA)/2sinA(1 - sinA)
= (1 + cosA)/sinA
= cosecA + cotA = RHS
multiplying and dividing (cosA - sinA + 1)
= (cosA - sinA + 1) × (cosA - sinA + 1)/(cosA + sinA - 1 ) × (cosA - sinA + 1)
= (cosA - sinA + 1)²/{cos²A - (sinA - 1)²}
= (cos²A + sin²A - 2sinA.cosA + 1 + 2cosA - 2sinA)/{cos²A - sin²A - 1 + 2sinA}
= (1 - 2sinA.cosA + 1 + 2cosA - 2sinA)/(1 - sin²A - sin²A - 1 + 2sinA)
= (2 - 2sinA + 2cosA - 2sinA.cosA)/(2sinA - 2sin²A)
= {2(1 - sinA) + 2cosA(1 - sinA)}/{2sinA(1 - sinA)}
= 2(1 + cosA)(1 - sinA)/2sinA(1 - sinA)
= (1 + cosA)/sinA
= cosecA + cotA = RHS
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